When computing the Fourier transformation of a mathematical function or of the waveform representing an electrical signal it is necessary to select the number of “points” or locations in the waveform at which the Fourier transformation coefficients will be determined. Alternately this selection may be expressed as a need to determine in advance the number of terms to be included in the Fourier series used to represent the waveform or the signal in the transform output. Thus it is common practice to speak of for example a ten point or a one hundred point or a two hundred fifty six point Fourier transformation. Each such point of the achieved Fourier transformation includes a coefficient magnitude for a component frequency in the sought-after Fourier series representing the mathematical function or the waveform.
The Fourier transformation accomplished in this manner has in effect performed the function of filtering the input signal of the receiver or other apparatus employing the Fourier transformation operation into a number of frequency components or frequency bins of predetermined frequency location. The Fourier series summation of these frequency components or frequency bins is of course a representation of the original input signal in the frequency domain, a representation made up of components having the selected frequencies of the frequency bins. In the present invention the relationship between an incoming signal frequency and the selected location of these frequency bins is considered using the example of a global position system receiver.
The expression “frequency bin” is widely used in referring to the terms of a Fourier transformation series. In keeping with this practice, and in extension thereof, the terms “frequency bin” and “frequency band” are employed in the present document in situations wherein it is helpful to segregate the results of the first and second Fourier transformation operations employed. Although this practice essentially recognizes generic and specific meanings for the term “frequency bin” it is believed that adjacent language provides clarifying segregation in each instance herein.
When discrete Fourier transform (DFT) or fast Fourier transform (FFT) methods are used for global position system signal or other signal acquisition purposes, it is found that if the input signal frequency happens to be located “on” the frequency of a Fourier transformation term, i.e., on a frequency bin frequency, then the amplitude of the acquired signal output from the Fourier transformation has a maximum value. When the acquired frequency is displaced from a frequency bin location however (since the input signal may occur at any frequency in the spectrum under consideration and since the Fourier transformation frequencies are fixed once the algorithm is implemented) the acquired signal may generate output responses in more than one nearby frequency bin however each such response is diminished in magnitude with respect to what it would have been if located on a frequency bin frequency. In such instances the Fourier transformation output magnitude is decreased or attenuated, possibly to an undesirable or intolerable degree. Such attenuation is especially undesirable in a signal acquisition situation where it may cause an already weak signal to remain lost in a noise background. For the sake of identification and easy referral this input frequency to Fourier transformation frequency difference difficulty is herein referred-to by the name of “frequency offset signal attenuation” or similar names. The present invention addresses this difficulty.
When an input signal is located midway intermediate two frequency bins a worst-case frequency offset attenuation situation of interest in the present invention exists. When this mid way frequency relationship occurs, the most affected Fourier transformation component incurs a 3.92 dB (i.e., 20×log(0.6366)) loss in amplitude. This amplitude loss may also be described as a receiver sensitivity loss, a loss of the same 3.92 dB in magnitude. It is often highly desirable to recover some of this loss in order to achieve desirable GPS receiver system performance. The present invention is believed to provide a desirable resolution of this difficulty.